New Steffensen Pairs

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New Steffensen Pairs

This is the Published version of the following publication

Qi, Feng and Cheng, Jun-Xiang (2000) New Steffensen Pairs. RGMIA research report collection, 3 (3).

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FENG QI AND JUN-XIANG CHENG

Abstract. In this article, using mathematical induction and analytic techniques, some new Steffensen pairs are established.

1. Introduction

Let f and g be integrable functions on [a, b] such thatf is decreasing and 0 6 g(x) 61 for x∈[a, b]. Then

Z b b−λ

f(x)dx6Z b a

f(x)g(x)dx6Z a+λ a

f(x)dx, (1)

whereλ=Rb

ag(x)dx.

The inequality (1) is called Steffensen’s inequality. For more information, please see [3, 4, 14].

In [1], its discrete analogue of inequality (1) was proved: Let {xi}ⁿi=1 be a decreasing finite sequence of nonnegative real numbers, {yi}ⁿi=1 be a finite sequence of real numbers such that 06yi61 for 16i6n. Letk1, k2∈ {1,2, . . . , n} be such thatk26Pⁿ

i=1

yi6k1. Then Xn

i=n−k2+1

xi6 Xn i=1

xiyi6

k1

X

i=1

xi. (2)

As a direct consequence of inequality (2), we have: Let{xi}ⁿi=1be nonnegative real numbers such that Pⁿ

i=1

xi 6Aand Pⁿ

i=1

x²_i >B², whereA andB are positive real numbers. Letk∈ {1,2, . . . , n} be such thatk>B^A. Then there areknumbers amongx1, x2, . . . , xn whose sum is bigger than or equals toB.

The so-called Steffensen pair was defined in [2] as follows:

Date: July 13, 2000.

1991Mathematics Subject Classification. Primary 26D15.

Key words and phrases. Steffensen inequality, Steffensen pair, generalized weighted mean values, extended mean values, mathematical induction, absolutely monotonic function.

The authors were supported in part by NSF of Henan Province (no. 004051800), SF for Pure Research of the Education Committee of Henan Province (no. 1999110004), and Doctor Fund of Jiaozuo Institute of Technology, The People’s Republic of China.

1

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2 FENG QI AND JUN-XIANG CHENG

Definition 1. Letϕ: [c,∞)→[0,∞) andτ : (0,∞)→(0,∞) be two strictly increasing functions, c >0, let {xi}ⁿi=1 be a finite sequence of real numbers such that xi >c for 1 6i 6n, A and B be positive real numbers, and

Pn i=1

xi6A, Pn i=1

ϕ(xi)>ϕ(B). If, for anyk∈ {1,2, . . . , n} such that k>τ(^A_B), there areknumbers amongx1, . . . , xn whose sum is not less thanB, then we call (ϕ, τ) a Steffensen pair on [c,∞).

In [1] and [2], the following Steffensen pairs were found respectively:

(x^α, x^1/(α⁻¹⁾), α>2, x∈[0,∞);

(xexp(x^α−1),(1 + lnx)^1/α), α>1, x∈[1,∞).

Letaandbbe real numbers satisfyingb > a >1 and√

ab>e. Define ϕ(x) =







x^1+ln^b−x^1+ln^a

lnx , x >1;

lnb−lna, x= 1, τ(x) =x^1/^ln^√^ab.

Then it was verified in [2] that (ϕ, τ) is a Steffensen pair on [1,∞).

In this article, we will establish some new Steffensen pairs, that is

Theorem 1. If aandb are real numbers satisfyingb > a >1 orb > a⁻¹>1, and√

ab>e, then

x Z b

a

t^ln^x⁻¹dt, x^1/^ln^√^ab

!

(3) is a Steffensen pair on [1,+∞).

If aandb are real numbers satisfyingb > a >1and√

ab>e, then

x Z b

a

(lnt)ⁿt^ln^x⁻¹dt, x

n+2

n+1·^(ln_(ln^b)_b)ⁿ⁺¹n+2⁻−^(lna)(lna)ⁿ⁺¹n+2

!

(4) are Steffensen pairs on[1,+∞)for any positive integer n.

Remark 1. This theorem generalizes the Proposition 2 in [2].

2. Lemmas

Lemma 1([2]). Let ψ: [c,∞)→[0,∞)be increasing and convex, c>0. Assume thatψ satisfies ψ(xy)>ψ(x)g(y) for all x>c and y >1, where g : [1,∞)→ [0,∞) is strictly increasing. Set ϕ(x) =xψ(x),τ(x) =g⁻¹(x), whereg⁻¹ is the inverse function ofg. Then (ϕ, τ) is a Steffensen pair on[c,∞).

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Letb > a >1 orb > a⁻¹>1, and√

ab > e. Define

h(x) =





 b^x−a^x

x , x6= 0;

lnb−lna, x= 0.

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It can be represented in integral form in [5]—[13] as follows h(x) =

Z b a

t^x⁻¹dt, x∈R. (6)

It had been verified in [10] that the function h(x) is absolutely and regularly monotonic on (−∞,+∞) forb > a >1, or on (0,+∞) forb > a⁻¹>1, completely and regularly monotonic on (−∞,+∞) for 0 < a < b <1, or on (−∞,0) for 1< b < a⁻¹. Furthermore, h(x) is absolutely convex on (−∞,+∞).

A function f(t) is said to be absolutely monotonic on (c, d) if it has derivatives of all orders and f^(k)(t) > 0 for t ∈ (c, d) and k ∈ N. For information of absolutely (completely, regularly, respectively) monotonic (convex, respectively) function, please refer to [4, 6, 10].

Lemma 2. Forx>0 andn>0, we have

h⁽ⁿ⁺¹⁾(x)>h⁽ⁿ⁾(x). (7)

Proof. It is clear that

h⁽ⁿ⁾(x) = Z b

a

t^x⁻¹(lnt)ⁿdt. (8)

By the Tchebysheff’s integral inequality or by Cauchy-Schwarz-Buniakowski inequality as in [5]—[13], we have

[h⁽ⁿ⁺¹⁾(x)]²6h⁽ⁿ⁾(x)h⁽ⁿ⁺²⁾(x). (9) Since the extended mean valuesE(r, s;u, v) defined in [5, 9, 12] by

E(r, s;u, v) =

r

s· u^s−v^s u^r−v^r

1/(s−r)

, rs(r−s)(u−v)6= 0; (10)

E(r,0;u, v) =

u^r−v^r lnu−lnv ·1

r

1/r

, r(v−u)6= 0; (11)

E(r, r;u, v) = e⁻^1/r

u^u^r v^v^r

1/(u^r−v^r)

, r(u−v)6= 0; (12)

E(0,0;u, v) =√

uv, u6=v;

E(r, s;u, u) =u, u=v;

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are increasing withrandsfor fixed positive numbersuandv, then, for everyy>0, the function F(x) = ^h(x+y)_h(x) is increasing withx. Therefore

F⁰(x) =h⁰(x+y)h(x)−h(x+y)h⁰(x) [h(x)]² >0, hence

h⁰(x+y)h(x)−h(x+y)h⁰(x)>0 (13) holds for allxandy>0.

Takingx= 0 in (13), for ally>0, we obtain

h⁰(y)h(0)−h(y)h⁰(0)>0. (14)

Sinceh⁰(0) =h(0) ln√

ab and√

ab>e, we haveh⁰(0)>h(0), and h⁰(y)>h(y) fory>0.

Note that inequality (13) can also be obtained from Lemma 4 in [12]: The functions ^h^(2(k+i)+1)_h(2k)(t)^(t)

are increasing with respect tot foriandkbeing nonnegative integers.

By mathematical induction, assume thath⁽ⁿ⁺¹⁾(x)>h⁽ⁿ⁾(x) forn >1 andx>0. Then, from inequality (9), we obtain

h⁽ⁿ⁾(x)h⁽ⁿ⁺¹⁾(x)6[h⁽ⁿ⁺¹⁾(x)]²6h⁽ⁿ⁾(x)h⁽ⁿ⁺²⁾(x), (15) therefore

h⁽ⁿ⁺¹⁾(x)6h⁽ⁿ⁺²⁾(x).

The proof is completed.

3. Proof of Theorem 1 Now we give a proof of Theorem 1.

Setψ(x) =h⁽ⁿ⁾(lnx) forx>1 andn>0. Direct computation yields thatψ⁰(x) = ^h⁽ⁿ⁺¹⁾_x^(ln^x)>

0 andψ⁰⁰(x) =^h⁽ⁿ⁺²⁾^(lnx)_x⁻2^h⁽ⁿ⁺¹⁾^(ln^x)>0. Henceψ(x) is increasing and convex.

Let u, v, r, s ∈ R, let p 6≡ 0 be a nonnegative and integrable function and f a positive and integrable function on the interval between xandy. Then the generalized weighted mean values Mp,f(r, s;u, v) of the functionf with weight pand two parametersrandsare defined in [6] by

Mp,f(r, s;u, v) = Rv

up(t)f^s(t)dt Rv

up(t)f^r(t)dt

!1/(s−r)

, (r−s)(u−v)6= 0; (16)

Mp,f(r, r;u, v) = exp Rv

u p(t)f^r(t) lnf(t)dt Rv

up(t)f^r(t)dt

!

, u−v6= 0; (17)

M(r, s;u, u) =f(u).

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¿From the Cauchy-Schwarz-Buniakowski inequality and standard argument, it was obtained in [13] that: The generalized weighted mean values Mp,f(r, s;u, v) are increasing with both r and s for any given continuous nonnegative weight p, continuous positive function f, and fixed real numbersuandv. Then, ifb > a >1, forx, y>0 andn>1, we have

h⁽ⁿ⁾(x+y) h⁽ⁿ⁾(x) =

Rb

a t^x+y⁻¹(lnt)ⁿdt Rb

at^x⁻¹(lnt)ⁿdt >exp

y· n+ 1

n+ 2· (lnb)ⁿ⁺²−(lna)ⁿ⁺² (lnb)ⁿ⁺¹−(lna)ⁿ⁺¹

. (18)

Therefore, forx, y>1, ψ(xy)

ψ(x) = h⁽ⁿ⁾(ln(xy))

h⁽ⁿ⁾(lnx) =h⁽ⁿ⁾(lnx+ lny) h⁽ⁿ⁾(lnx) >y

n+1

n+2·^(ln_(ln^b)_b)ⁿ⁺²n+1⁻−^(lna)(lna)ⁿ⁺²n+1. (19) Letg(x) =xⁿ⁺¹ⁿ⁺²^·

(lnb)n+2−(lna)n+2

(lnb)n+1−(lna)n+1 for x>1, then g⁻¹(x) =xⁿ⁺²ⁿ⁺¹^·

(lnb)n+1−(lna)n+1

(lnb)n+2−(lna)n+2, x∈[1,+∞).

By Lemma 1, (ϕ, τ), where ϕ(x) = xψ(x) = xh⁽ⁿ⁾(lnx) = xRb

a(lnt)ⁿt^ln^x⁻¹dt and τ(x) = x

n+2

n+1·^(lnb)_(lnb)ⁿ⁺¹n+2⁻−^(ln(ln^a)a)ⁿ⁺¹n+2 forx>1 andn>0, are Steffensen pairs on [1,+∞) for any givenn>0.

Ifaandbare real numbers satisfyingb > a >1 orb > a⁻¹>1, and√

ab>e, then, forx, y>0, we have

h(x+y) h(x) >√

aby

. (20)

Therefore, forx, y>1,

ψ(xy)

ψ(x) =h(ln(xy))

h(lnx) =h(lnx+ lny)

h(lnx) >y^ln^√^ab. (21)

Letg(x) =x^ln^√^ab forx>1, theng⁻¹(x) =x^1/ln^√^ab, x∈[1,+∞). By Lemma 1, (ϕ, τ), where ϕ(x) =xψ(x) = xh(lnx) =xRb

a t^ln^x⁻¹dt and τ(x) =x^1/ln^√^ab forx>1, is a Steffensen pair on [1,+∞).

The proof is complete.

Remark 2. If considering the function Ry

x p(u)f^t(u)du, then more new Steffensen pairs can be obtained. We will discuss this in a subsequent paper [8].

References

[1] J.-C. Evard and H. Gauchman,Steffensen type inequalities over general measure spaces, Analysis17(1997), 301-322.

[2] H. Gauchman,Steffensen pairs and associated inequalities, Journal of Inequalities and Applications5(2000), no. 1, 53–61

[3] D. S. Mitrinovi´c,Analytic Inequalities, Springer-Verlag, Berlin, 1970.

[4] D. S. Mitrinovi´c, J. E. Peˇcaric and A. M. Fink,Classical and New Inequalities in Analysis, Kluwer Academic Publishers, 1993.

[5] Josip Peˇcari´c, Feng Qi, V. ˇSimi´c, and Sen-Lin Xu,Refinements and extensions of an inequality, III, Journal of Mathematical Analysis and Applications227(1998), no. 2, 439–448.

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[6] Feng Qi,Generalized weighted mean values with two parameters, Proceedings of the Royal Society of London Series A—Mathematical, Physical and Engineering Sciences454(1998), no. 1978, 2723–2732.

[7] Feng Qi,On a two-parameter family of nonhomogeneous mean values, Tamkang Journal of Mathematics29 (1998), no. 2, 155–163.

[8] Feng Qi and Jun-Xiang Cheng, On Steffensen pairs, RGMIA Research Report Collection 3 (2000).

http://rgmia.vu.edu.au.

[9] Feng Qi and Qiu-Ming Luo,A simple proof of monotonicity for extended mean values, Journal of Mathematical Analysis and Applications224(1998), 356–359.

[10] Feng Qi and Sen-Lin Xu,The function(b^x−a^x)/x: Inequalities and properties, Proceedings of the American Mathematical Society126(1998), 3355–3359.

[11] Feng Qi and Sen-Lin Xu,Refinements and extensions of an inequality, II, Journal of Mathematical Analysis and Applications211(1997), 616–620.

[12] Feng Qi, Sen-Lin Xu, and Lokenath Debnath,A new proof of monotonicity for extended mean values, Inter- national Journal of Mathematics and Mathematical Sciences22(1999), no. 2, 415–420.

[13] Feng Qi and Shi-Qin Zhang,Note on monotonicity of generalized weighted mean values, Proceedings of the Royal Society of London Series A—Mathematical, Physical and Engineering Sciences 455(1999), no. 1989, 3259–3260.

[14] J. F. Steffensen,On certain inequalities and methods of approximation, J. Inst. Actuaries51(1919), 274-297.

Department of Mathematics, Jiaozuo Institute of Technology, Jiaozuo City, Henan 454000, The People’s Republic of China

E-mail address:[email protected]

URL:http://rgmia.vu.edu.au/qi.html or http://rgmia.vu.edu.au/Members/FQi.htm

Department of Mathematics, Jiaozuo Institute of Technology, Jiaozuo City, Henan 454000, The People’s Republic of China